Arithmetic Sequences

At a previous job of mine, there was a policy of holding a dinner party for the company each time we hired a new employee. At these dinners, each employee was treated to a $20 dinner at the expense of the company. There was also a manager responsible for keeping track of the costs of these dinners.

In computing the costs, the manager noticed that each time there is a new dinner, it was $20 more expensive than the last one. So if we let a1 represent the cost of the first dinner, and let ai represent the cost of the ith dinner, then we see that ai = ai-1 + 20. Sequences like this, where t arise quite often in practice and are called arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive numbers is constant.

For the example above, the term an will represent the cost of dinner after the nth employee has joined the company (assuming that no employees have left the company over this time period). Also the term Sn will represent the total cost the company has paid towards these dinners.

Before we continue with this example, consider the following table which lists the first five terms of an arithmetic sequence as well as the common difference and the first five sums of this sequence.

term number

term value

diff

sum number

sum value

a1

4

3

S1

4

a2

7

3

S2

11

a3

10

3

S3

21

a4

13

3

S4

34

a5

16

3

S5

50

One of the beauties of arithmetic sequences is that if we know the first term (a1) and the common difference (d), then we can easily calculate the terms an and Sn for any n with the following formulas:

We can use these formulas to derive more information about the sequence. For example, if my manager wanted to estimate the cost of dinners once we had added 30 new employees, this would be term a30 of the sequence, which we can evaluate with the above formula by a30 = a1 + d*(n – 1) = 0 + 20*(30 – 1) = 0 + 20 * 29 = 580.